Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 2005

Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 1905

Ruler & Compass Algorithms

Gauss: 17-gon

Constructing Regular N-gons 3folklore 5antiquity 17 Gauss (1796) 257 Richelot (1832) Hermes (1879) Gauss Fermat primes 2 2 k can’t do heptagons +1 proof covers a gym Hilbert proved lower bounds on number of steps

Tools from Computational Geometry Bernard Chazelle Princeton University Bernard Chazelle Princeton University Tutorial FOCS 2005

algorithmic algorithmic analytical analytical TOOLS FROM COMPUTATIONAL GEOMETRY TOOLS FROM COMPUTATIONAL GEOMETRY

1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

Voronoi Diagram

Works well also for convex hulls, nearest neighbors [3,6]

Works not so well for multidimensional searching: quadtrees, kd-trees: highly sub-optimal

Hopcroft’s problem Any point/line incidence?N points and N lines

Naïve divide & conquer

O(N logN) time Point location in line arrangement

~ O(N ) time 3/2

1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

1 Algorithmic tools 1 Algorithmic tools geometric divide & conquer geometric divide & conquer

[2, p.123]

N points

number of intersections = O( ) for any line

Often, the number of simple polygons is exponential.

Sometimes, it’s unique… Often, the number of simple polygons is exponential.

Sometimes, it’s unique… Often, the number of simple polygons is exponential.

N points number of intersections = O( ) SPANNING PATH THEOREM for any line

N points number of intersections = O( ) SPANNING PATH THEOREM for random line

Join two closest points; remove; repeat

number of intersections = O( ) for random line

Difficulty 1: Produces a matching, not a simple polygon

Matching  Tree

Remove each edge and one of its adjacent vertices

number of intersections = O( ) = O( ) for random line

Tree  Hamiltonian Circuit

(via DFS)

Hamiltonian Circuit  Simple Polygon

(via edge switching)

Simple Polygon

for random line number of intersections = O( )

Change definition of randomness

New definition: A random line joins 2 of the N points picked at random.

Next goal A random line cuts O( ) edges

Euclidean is wrong metric

prob [ line(random pair) cuts ab ] b a New metric: d(a,b)= pick random pair

New metric has “dimension” 2 b a pick random pair

This ensures that a random line cuts O( ) edges

Final goal: A line between any 2 points picked at cuts O( ) edges

Increase d(a,b) multiplicatively (as in BOOSTING ) a b double probability of picking pair

ANY line cuts O( ) edges

Spanning Path Theorem

APPLICATION: SIMPLEX RANGE COUNTING [2, p.214] How many points in the triangle? 6 6

Ray shooting in O(log N) time

APPLICATION: SIMPLEX RANGE COUNTING How many points in the triangle?

APPLICATION: SIMPLEX RANGE COUNTING Triangle range counting in O( ) time ~

Spanning Path Theorem

-APPROXIMATION (for triangles)

Subset A such that: any triangle T

Subset A such that: any triangle T

Subset A such that: any triangle T

Size of A is O( ) Independent of N Better than random! Size of A is O( ) Independent of N Better than random! -4/3 ~

Keep every other edge

Color randomly red/blue

discrepancy within any triangle = ?

discrepancy within any triangle = 1

discrepancy within any triangle =

Remove red points

Recolor

Remove red points

Repeat until O( ) points left ~ -4/3

Subset A such that: any triangle T

A is called an -approximation A is called an -approximation (for triangles) (for triangles) A is called an -approximation A is called an -approximation (for triangles) (for triangles) Its size O( ) is independent of N Its size O( ) is independent of N -4/3 ~ A is computable in poly(N) A is computable in poly(N)

Set System (X, ) Set System (X, ) 2 2 XX VC dim = max |shattered set|

VC dim = 3 VC dim = 3

Unbounded VC dimension

Bounded VC dim implies that Bounded VC dim implies that Given any Y X, number of distinct sets Y S, where S, is O(|Y| ) Given any Y X, number of distinct sets Y S, where S, is O(|Y| ) cc Dual set system  dual shatter exponent primal shatter exponent easy to determine

VC dim = ? VC dim = ? (points, ellipsoids) in d-dim (points, ellipsoids) in d-dim

dual shatter function = O(N ) dual shatter function = O(N ) (points, ellipsoids) in d-dim d by Thom-Milnor

Set System (V, S) Set System (V, S) O( ) -2+2/(d+1) ~ d= VC dimension Size of -approximation is or primal/dual shatter exponent [2, p.179]

Set System (V, S) Set System (V, S) O( ) -2 ~ Size of -approximation is Computable in O(N) poly( ) In comp geom, random bits help with simplicity but not with complexity [2, p.175]

-cutting -cutting N lines

[2, p.204]

Application Hopcroft’s problem [2, p.213]

Dualize point (a,b) line aX+bY=1

Recurse

Hopcroft’s problem

N lines Standard sampling Standard sampling How many lines?

Set System (X, ) Set System (X, ) X = set of N lines X = set of N lines = =

N lines easy to do with an -approximation How many lines?

N lines Product sampling Product sampling How many vertices? [2, p.183]

N lines Unbounded VC-dim: yet can be done! Unbounded VC-dim: yet can be done! How many vertices?

Convex hull of N points in R dd [ 2, p.283]

Voronoi diagram of N points in E dd

Linear programming in linear time with fixed number of variables LP-type programming in linear time with fixed number of variables [1, p.82]

Linear programming in linear time with fixed number of variables LP-type programming in linear time with fixed number of variables [2, p.307]

d Smallest ellipsoid enclosing N points in R [2, p.313] in O (N) time! d

[0] Sampling tool for approximate geometric optimization

2 Analytical tools 2 Analytical tools 2.1 randomized scaling 2.2 backward analysis 2.1 randomized scaling 2.2 backward analysis

2 Analytical tools 2 Analytical tools 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis

2 Analytical tools 2 Analytical tools 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis

K-SETS

n(i,j) = 9 p p i j f = { (i,j) | i<j and n(i,j)= k } k

f = 6 0

Theorem:Theorem: [4, p.141]

X = 3

Theorem:Theorem:

Theorem:Theorem:

2 Analytical tools 2 Analytical tools 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis

The Crossing Lemma: [4, p.55]

Pick each vertex with prob p Set p= 4n/m

Corollary:Corollary: # point/line incidences = O(N ) 4/34/3

Corollary:Corollary: # unit-distance pairs = O(N ) 4/34/3

2 Analytical tools 2 Analytical tools 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis 2.1 randomized scaling k-sets crossing lemma 2.2 backward analysis

Linear Programming Linear Programming [1, p76]

N constraints and d variables

Planar Graph

Planar Separator Theorem Remove O( ) vertices  (1/3-2/3) cut

[5, p96]

Stereographic lifting

Centerpoint Theorem (1/3,2/3) cut (1/4,3/4) cut in 3D

Can assume centerpoint is center of sphere

BIBLIOGRAPHYBIBLIOGRAPHY The results mentioned in this tutorial, as well as the history behind them, are discussed in detail in the surveys and monographs below.